Theorem cbvsbdavw2 36213

Description: Change bound variable in proper substitution. General version of cbvsbdavw 36212. Deduction form. (Contributed by GG, 14-Aug-2025.)

Hypotheses

Ref	Expression
cbvsbdavw2.1	⊢ (𝜑 → 𝑧 = 𝑤)
cbvsbdavw2.2	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
cbvsbdavw2	⊢ (𝜑 → ([𝑧 / 𝑥]𝜓 ↔ [𝑤 / 𝑦]𝜒))

Distinct variable groups:   𝜑,𝑥,𝑦   𝜓,𝑦   𝜒,𝑥

Allowed substitution hints:   𝜑(𝑧,𝑤)   𝜓(𝑥,𝑧,𝑤)   𝜒(𝑦,𝑧,𝑤)

Proof of Theorem cbvsbdavw2

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvsbdavw2.1	. . . . 5 ⊢ (𝜑 → 𝑧 = 𝑤)
2		equequ2 2025	. . . . 5 ⊢ (𝑧 = 𝑤 → (𝑡 = 𝑧 ↔ 𝑡 = 𝑤))
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → (𝑡 = 𝑧 ↔ 𝑡 = 𝑤))
4		equequ1 2024	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑡 ↔ 𝑦 = 𝑡))
5	4	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑥 = 𝑡 ↔ 𝑦 = 𝑡))
6		cbvsbdavw2.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
7	5, 6	imbi12d 344	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → ((𝑥 = 𝑡 → 𝜓) ↔ (𝑦 = 𝑡 → 𝜒)))
8	7	cbvaldvaw 2037	. . . 4 ⊢ (𝜑 → (∀𝑥(𝑥 = 𝑡 → 𝜓) ↔ ∀𝑦(𝑦 = 𝑡 → 𝜒)))
9	3, 8	imbi12d 344	. . 3 ⊢ (𝜑 → ((𝑡 = 𝑧 → ∀𝑥(𝑥 = 𝑡 → 𝜓)) ↔ (𝑡 = 𝑤 → ∀𝑦(𝑦 = 𝑡 → 𝜒))))
10	9	albidv 1919	. 2 ⊢ (𝜑 → (∀𝑡(𝑡 = 𝑧 → ∀𝑥(𝑥 = 𝑡 → 𝜓)) ↔ ∀𝑡(𝑡 = 𝑤 → ∀𝑦(𝑦 = 𝑡 → 𝜒))))
11		df-sb 2065	. 2 ⊢ ([𝑧 / 𝑥]𝜓 ↔ ∀𝑡(𝑡 = 𝑧 → ∀𝑥(𝑥 = 𝑡 → 𝜓)))
12		df-sb 2065	. 2 ⊢ ([𝑤 / 𝑦]𝜒 ↔ ∀𝑡(𝑡 = 𝑤 → ∀𝑦(𝑦 = 𝑡 → 𝜒)))
13	10, 11, 12	3bitr4g 314	1 ⊢ (𝜑 → ([𝑧 / 𝑥]𝜓 ↔ [𝑤 / 𝑦]𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535  [wsb 2064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-sb 2065

This theorem is referenced by: (None)